Proof of Theorem 1:
Proof of Theorem 2:This is a modification of a proof for a version of the theorem given in Troutt (1993). By the assumption that x is uniformly distributed on {x:w(x) = u}, f(x) must be constant on these contours; so that f(x) = φ(w(x)) for some function, φ( ). Consider the probability P( u ≤ w(x) ≤ u + ε ) for a small positive number, ε. On the one hand, this probability is ε g(u) to a first order approximation. On the other hand, it is also given by
Therefore
Division by ε and passage to the limit as ε → 0 yields the result.
Further Details on the Simulation ExperimentTo simulate observations within each data set, a uniform random number was used to choose between the degenerate and continuous portions in the density model
where p =0.048, α =1.07, and β =0.32. With probability p, δ(0) was chosen and w =0 was returned. With probability 1p, the gamma (α,β) density was chosen and a value, w, was returned using the procedure of Schmeiser and Lal (1980) in the IMSL routine RNGAM. The returned w was converted to an efficiency score, v, according to v =exp{−w^{0.5}}. For each v, a vector Y was generated on the convex polytope with extreme points e_{1} =(v/a_{1}*,0,0,0), e_{2} =(0,v/a_{2}*,0,0), e_{3} =(0,0,v/a_{3}*,0) and e_{4} =(0,0,0,v/a_{4}*) using the method given in Devroye (1986).
 
